This is the user sandbox of Scwarebang. A user sandbox is a subpage of the user's user page. It serves as a testing spot and page development space for the user and is not an encyclopedia article. Create or edit your own sandbox here. Other sandboxes: Main sandbox | Template sandbox Finished writing a draft article? Are you ready to request review of it by an experienced editor for possible inclusion in Wikipedia? Submit your draft for review!

/info/en/?search=User:Scwarebang/Books/uncertain_reals (1958-08-27) August 27, 1958 (age 65)

The Institute for Risk and Uncertainty (commonly known as the "Risk Institute") is a component of the Faculty of Science and Engineering of the University of Liverpool in the United Kingdom. The Risk Institute conducts internationally recognised research and training in methods and tools to manage the inherent risks and inescapable uncertainties that arise in natural, social and engineered systems across all academic disciplines and fields of endeavour.

The Institute is funded primarily by research and training grants from UKRI, but it collaborates with researchers, academic institutions, government agencies, and industrial concerns from around the world. Collaboration with industry and NGOs is central to its research and training.

The Institute hosts the Centre for Doctoral Training in Quantification and Management of Risk & Uncertainty in Complex Systems & Environments with funding from the EPSRC and ESRC, which is collocated with the new EPSRC Centre for Doctoral Training in Distributed Algorithms ^[1].

The Risk Institute's mission ^[2] is to create methods to quantify, mitigate and manage risk and uncertainty to help people and organisations create a safer, more secure, and more efficient world.

where natural and engineered systems can exhibit extreme or unfavourable states that can lead to injury or death, financial loss, or just suboptimal performance. Anticipating and preventing these outcomes requires understanding how they arise given fluctuations in environmental conditions, actions by adversaries, imperfect or limited measurements, and incomplete scientific understanding of the underlying physical processes.

The Risk Institute promotes risk analysis and uncertainty quantification as a part of science and engineering that uses knowledge from physics, biology, chemistry, environmental and life sciences, medicine, economics and finance, psychology and social sciences for solving diverse problems. It develops new methods, experimental and numerical tools, products, and technological and service innovations for engineering and mathematical modeling of the natural and social resources.

Research and development at the Risk Institute is both inter- and trans-disciplinary, spaning four broad areas: ^[3]

• Measurement and uncertainty characterisation to make proper use of the statistical uncertainty in measurements collected in new or fluctuating environments or when imprecision cannot be neglected,
• Numerical simulation methods to integrate available data and incomplete scientific knowledge about the underlying processes into models to forecast relevant risks,
• Risk and uncertainty communication to overcome recognised biases afflicting human perception and cognition involving uncertainties, and
• Decision making under risk and uncertainties to optimise planning and management of complex systems.

In response to a diverse series of dramatic disasters in the first decade of the century ^[4], the Risk Institute was founded in 2012 by Professor Michael Beer, who was its director until 2015. ^[5]

• Institute for Risk and Uncertainty
• University of Liverpool

Category:Research institutes in the UK Category:Research institutes established in 2011

THIS WAS THE START OF THE COMMENTING

In the theory of estimation, bounding is a generic strategy to identify or numerically localize a quantity in the face of uncertainty. Bounding estimates quantities from the outside inward. Bounding can be contrasted with approximation which does not control the direction of the approach. Whereas approximations are concerned with the size of the approximation error, i.e., the difference between the estimate and the true value of the quantity or quantities being estimated, bounding is concerned with the tightness of the estimation about the true value(s). Two kinds of bounds are commonly discussed: rigorous and statistical. Rigorous bounds on a quantity represent a lower limit and an upper limit beyond which analysts are sure—or assume—that the quantity cannot fall. Statistical bounds, perhaps in the form of a frequentist or Bayesian confidence interval, are nonrigorous in that there is no guarantee or presumption that the corresponding true value necessarily falls within the bounds, only that the confidence or probability that it does is at least as large as some prescribed level. Bounds that are neither rigorous nor statistical are essentially lower and upper approximations.

Copula_(probability_theory)#Fr.C3.A9chet.E2.80.93Hoeffding_copula_bounds

In probability theory, Fréchet bounds, sometimes called Fréchet–Hoeffding bounds, are the optimal bounds on a joint or multivariate distribution function given only its marginal univariate distribution functions, without specifying the copula or dependence function that knits the marginals together. For instance, a bivariate distribution function H(x,y) = P(X ≤ x, Y ≤ y) describing random variables X and Y is sure to satisfy the inequality

max{F(x) + G(y) − 1, 0} ≤ H(x,y) ≤ min{F(x),G(y)}, for all real numbers x, y

where F(x) = P(X ≤ x) and G(y) = P(Y ≤ y) are the marginal distribution functions of H.

These bounds are related to the Fréchet inequalities which are analogous bounds on the probabilities of joint events.

They are also related to dependency bounds ^[1] which are bounds on the cumulative distribution functions of random variables when only the marginal distributions of the random variables are known, and nothing about their stochastic dependence.

Sklar

Let Ω be some universe, space or set of interest. Often, this is either the real numbers ℝ or the integers ℤ, but it could be any topological space. <<Does it even need a topology? It doesn’t need to be a metric space, does it? Doesn’t it at least need to be Hausdorff? Is separability an issue?>> For this discussion, let A, B, and C denote subsets of Ω. Suppose A ⊂ Ω is a set that we are trying to characterize.

Bounding can be a means of estimating quantities in the face of uncertainties, and as such it can be contrasted with approximation. .

Bounding can be considered to be a kind of outer approximating. Approximating is a way to identify or localize something, but without controlling the direction of the approach.

Rowe ^[1] pointed out there are a variety of reasons that bounding may be more convenient than an approximation, including the fact that rigorous bounds are

• usually relatively easy to compute compared to approximations which may involve integrals,
• very simple to combine via intersection,
• often sufficient to specify a decision,
• obtainable even in situations where estimates are impossible, and
• valuable because they are rigorous rather than approximative.

Exactly what is being bounded can make a substantial difference. For instance, the imprecise expression "France" can refer to somewhere in France as a fixed location whose actual position is not precisely known, or to anywhere in France as a dynamical range of a varying location that must at some point include every spot in France, or perhaps to the whole of France as the set of all locations in France at once and considered as one object. These interpretational issues are on top of the basal semantic imprecision in the word, as France may refer to Metropolitan France, Mainland France (the Hexagon), the modern French Republic, the country or its government at different points in history, or even the historical monarchs of France. Indeed, there are other possible meanings such as people, ships, or other entities with that name.

Let Ω be a set. Often, this set is either the reals or the integers, but it could be any topological space. <<Does it even need a topology? It doesn't need to be a metric space? Doesn't it at least need to be Hausdorff?>> Suppose A ⊂ Ω is the set of all solutions in Ω that we are trying to characterize.

Let A, B, and C denote subsets of Ω.

Let ω, θ, and φ denote elements of Ω.

If ω ∈ A, we call ω a solution. If ω ∉ A, we say ω is not a solution. If θ ∈ B, we say θ is implied by B.

If ω ∈ A, we call ω a solution. If ω ∉ A, we say ω is not a solution. If θ ∈ B, we say θ is implied by B.

We say that B ⊂ Ω bounds A (and that A is bounded by B) if every element of A is also an element of B. If B bounds A then

A ⊂ B,

A ⊆ B, or

A = B.

We say that B is an inner bound for A if A bounds B. We say that C constrains A if either

C bounds A or

A bounds C,

and that we know which of these two statements is true.

Structured bounds are less resolved than arbitrary sets but often far simpler to work with. Call D a structured bounding space for Ω if every subset of Ω is bounded by some element of D, i.e., if the following conditions hold:

D ⊆ 2^Ω, and

A ⊆ Ω implies A ⊆ B for some B ∈ D.

This definition implies that every element of Ω is covered by some element of D, and also that Ω itself is covered by some element of D which would be called a vacuous bound. Note that the empty set may or may not be an element of D.

The union of the elements in any subset of D is bounded by some element of D?

The bound B ∈ D is said to be a best-possible (relative to D) bound on A ⊆ Ω if both

A ⊆ B, and

there exists no C ∈ D such that A ⊆ C ⊂ B.

It may be further true that B is the unique best-possible bound in the sense that

A ⊆ C implies that B ⊆ C, for all C ∈ D.

Thus, B is the unique best-possible bound on A if B bounds A and it is the tightest bound in D that does so.

Some structured bounding spaces, such as intervals bounding sets of reals, have only unique best-possible bounds. Their elements form a tree-shaped upper lattice.

If Ω is the real numbers, then 2^ℝ then is a trivial structured bounding space (i.e., one with no structure) because no subsets of Ω are missing. The set of traditional intervals ^[2]

{ [a, b] : a ≤ b, a ∈ ℝ, b ∈ ℝ }

is not a structured bounding space for ℝ, because there is no interval that bounds the whole of ℝ itself. Likewise all half-bounded intervals of the form [a, ∞] and [−∞, b] are also not bounded by any element of the set of real intervals.

However, a structured bounding space for ℝ can be constructed as

{ [a, b] : a ≤ b, a ∈ ℝ*, b ∈ ℝ* }

where ℝ* = ℝ ∪ {−∞, ∞}.

A finite-precision number is

A significant-digit interval is the set of real numbers that are d-close to a real number e whose

Are significant-digit intervals a structured bounding space for the reals? SCOTT SAYS NO!

Are significant-digit intervals a structured bounding space for the reals? Are significant-digit intervals a structured bounding space for the reals? Are significant-digit intervals ℝ a structured bounding space for the reals? Are significant-digit intervals a structured bounding space for the reals? Are significant-digit intervals a structured bounding space for the reals?

There are various notation systems in use for denoting open, closed and partially open intervals.

(a,b) = ]a,b[ = [a+, b-] = [a plus, b minus]
(a,b] = ]a,b] = [a+, b]  = [a plus, b]
[a,b) = [a,b[ = [a, b-]  = [a, b minus]
[a,b] = [a,b] = [a, b]   = [a, b]

The system using both parentheses and brackets is conventional, but is often confused with ordered sets. The backward-bracket notation ]a,b[ was introduced by Bourbaki ^[3] is efficient, but it is easy to misread. Notation should strive to be helpful in addition to being efficient. When something should be noticed, the notation should help readers notice it. Thus, we prefer the plus-minus notation, which cannot be misread and, arguably, is the most intuitive of the three.

The plus-minus notation is more flexible, as it can designate a range which is a halo around a closed interval, for instance [a minus, b plus].

Material on kinds of bounds: socks sleeves mittens gloves

Simultaneous (distributional) confidence bounds versus point-wise confidence bounds

Algebraic structure of bounding and uncertainty calculi See /info/en/?search=Magma_(algebra)#Types_of_magma

An algebra A over a set Ω is a non-empty set of sets of Ω that satisfies: (i) Ω∈A, (ii) if a∈A then a^c∈A, and (iii) if a,b∈A then a∪b∈A. Thus an algebra is a set containing Ω that is closed under complements and finite unions and intersections.

A σ-algebra is a non-empty set of sets that is closed under complements and countable unions and intersections.

A measurable space (Ω,F) consists of a set Ω and a σ-algebra of subsets of Ω.